The Mathematics of GeoEnergy

Compartmental Models

Chapter 5 of the book describes a model of the production of oil based on discoveries followed by a sequence of lags relating to decisions made and physical constraints governing the flow of that oil. As it turns out, this so-named Oil Shock Model is mathematically similar to the compartmental models used to model contagion growth in epidemiology, pharmaceutical/drug deliver systems, and other applications as demonstrated in Appendix E of the book.

One aspect of the 2020 pandemic is that everyone with any math acumen is becoming aware of contagion models such as the SIR compartmental model, where S I R stands for Susceptible, Infectious, and Recovered individuals. The Infectious part of the time progression within a population resembles a bell curve that peaks at a particular point indicating maximum contagiousness. The hope is that this either peaks quickly or that it doesn’t peak at too high a level.

One compartmental population growth model, that specified by the Lotka-Volterra-type predator-prey equations, can be manipulated to match a cyclic wildlife population in a fashion approximating that of observations. The cyclic variation is typically explained as a nonlinear resonance period arising from the competition between the predators and their prey. However, a more realistic model may take into account seasonal and climate variations that control populations directly. The following is a recent paper by wildlife ecologist H. L. Archibald who has long been working on the thesis that seasonal/tidal cycles play a role (one paper that he wrote on the topic dates back to 1977! ).